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Özlem Imamoglu

Publications and source records attributed to Özlem Imamoglu.

10 recordsLinked to original sources
arXiv2025

Real part of cycle integrals and conjectures of Kaneko

We prove two of Kaneko's conjectures on the "values" $\mathrm{val}(w)$ of the modular $j$ function at real quadratic irrationalities: we prove the lower bound $\mathrm{Re}(\mathrm{val}(w))\geq \mathrm{val}\left(\frac{1+\sqrt{5}}{2}\right)$ for all real quadratics $w$ and the upper bound $\mathrm{Re}(\mathrm{val}(w))\leq \mathrm{val}\left(1+\sqrt{2}\right)$ for all Markov irrationalities $w$. These results generalize to the "values" at quadratic irrationalities of any weakly holomorphic modular function $f$ such that $f(e^{it})$ is real, non-negative and increasing for $t\in [π/3,π/2]$.

math.NT↗
arXiv2025

A Lyapunov exponent attached to modular functions

To each weakly holomorphic modular function $f\not \equiv 0$ for $\mathrm{SL}(2,\mathbb{Z})$, which is non-negative on the geodesic arc $\{e^{it} : π/3\leq t\leq 2π/3\}$, we attach a $\mathrm{GL}(2,\mathbb{Z})$-invariant map $Λ_f:\mathbb{P}^1(\mathbb{R})\to \mathbb{R}$ that generalizes the Lyapunov exponent function introduced by Spalding and Veselov. We prove that it takes every value between $0$ and $Λ_f\left(\frac{1+\sqrt{5}}{2}\right)$ and it gives an increasing convex function on the Markov irrationalities when ordered using their parametrization by Farey fractions in $[0,1/2]$. In the case of quadratic irrationals $w$ with purely periodic continued fraction expansion, the value $Λ_f(w)$ equals the real part of the cycle integral of $f$ along the associated geodesic $C_w$ on the modular surface, normalized with the word length of the associated hyperbolic matrix $A_w$ as a word in the generators $T=\left(\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right)$ and $V=\left(\begin{smallmatrix} 1 & 0 \\ 1 & 1 \end{smallmatrix}\right)$. These results are related to conjectures of Kaneko who observed several similar behavior for the cycle integrals of the modular $j$ function when normalized by the hyperbolic length of the geodesic $C_w$.

math.NT↗
arXiv2024

Twisted traces of modular functions on hyperbolic $3$-space

We compute analogues of twisted traces of CM values of harmonic modular functions on hyperbolic $3$-space and show that they are essentially given by Fourier coefficients of the $j$-invariant. From this we deduce that the twisted traces of these harmonic modular functions are integers. Additionally, we compute the twisted traces of Eisenstein series on hyperbolic $3$-space in terms of Dirichlet $L$-functions and divisor sums.

math.NT↗
arXiv2024

Special values of Green's functions on hyperbolic $3$-space

Gross, Kohnen and Zagier proved an averaged version of the algebraicity conjecture for special values of higher Green's functions on modular curves. In this work, we study an analogous problem for special values of Green's functions on hyperbolic $3$-space. We prove that their averages can be computed in terms of logarithms of primes and logarithms of units in real quadratic fields. Moreover, we study twisted averages of special values of Green's functions, which yield algebraic numbers instead of logarithms.

math.NT↗
arXiv2017

Modularity of generating series of winding numbers

The Shimura correspondence connects modular forms of integral weights and half-integral weights. One of the directions is realized by the Shintani lift, where the inputs are holomorphic differentials and the outputs are holomorphic modular forms of half-integral weight. In this article, we generalize this lift to differentials of the third kind. As an application we obtain a modularity result concerning the generating series of winding numbers of closed geodesics on the modular curve.

math.NT↗
arXiv2014

Holomorphic projections and Ramanujan's mock theta functions

We employ spectral methods of automorphic forms to establish a holomorphic projection operator for tensor products of vector-valued harmonic weak Maass forms and vector-valued modular forms. We apply this operator to discover simple recursions for Fourier series coefficients of Ramanujan's mock theta functions.

math.NT↗
arXiv2013

Estimates on the zeros of $E_2$

We improve the results from [EBS10] about the Eisenstein series $E_2(z)=1-24\sum_{n=1}^\infty \frac{nq^n}{1-q^n}$. In particular we show that there exists exactly one (simple) zero in each Ford circle and give an approximation to its location.

math.NT↗
arXiv2008

Palindromic random trigonometric polynomials

We show that if a real trigonometric polynomial has few real roots, then the trigonometric polynomial obtained by writing the coefficients in reverse order must have many real roots. This is used to show that a class of random trigonometric polynomials has, on average, many real roots. In the case that the coefficients of a real trigonometric polynomial are independently and identically distributed, but with no other assumptions on the distribution, the expected fraction of real zeros is at least one-half. This result is best possible.

math.PR↗
arXiv2008

Parabolic, hyperbolic and elliptic Poincaré series

Following Petersson, we study the parabolic, hyperbolic and elliptic expansions of holomorphic cusp forms and the associated Poincaré series. We show how these ideas extend to the space of second-order cusp forms.

math.NT↗